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Definition

There are a few definitions of the quotient topology however they do not conflict. This page might change shape while things are put in place

Quotient topology via an equivalence-relation definition

Given a topological space, $(X,\mathcal{J})$ and an equivalence relation on $X$ , $\sim$ ^{[Note 1]}, the quotient topology on $\frac{X}{\sim}$ , $\mathcal{K}$ is defined as:

The set K⊆P(X∼) such that:
- $\forall U\in\mathcal{P}(\frac{X}{\sim})[U\in\mathcal{K}\iff \pi^{-1}(U)\in\mathcal{J}]$ or equivalently
$\mathcal{K}=\{U\in\mathcal{P}(\frac{X}{\sim})\ \vert\ \pi^{-1}(U)\in\mathcal{J}\}$

In words:

The topology on $\frac{X}{\sim}$ consists of all those sets whose pre-image (under $\pi$ ) are open in $X$

Quotient topology via a mapping to a set definition

Let $(X,\mathcal{J})$ be a topological space and let $h:X\rightarrow Y$ be a surjective map onto a set $Y$ , then the quotient topology, $\mathcal{K}\subseteq\mathcal{P}(Y)$ is a topology we define on $Y$ as follows:

$\forall U\in\mathcal{P}(Y)[Y\in\mathcal{K}\iff h^{-1}(U)\in\mathcal{J}]$ or equivalently:
$\mathcal{K}=\{U\in\mathcal{P}(Y)\ \vert\ h^{-1}(U)\in\mathcal{J}\}$

The quotient topology on $Y$ consists of all those subsets of $Y$ whose pre-image (under $h$ ) is open in $X$

Claim 1: these definitions are equivalent

Universal property of the quotient topology

In this commutative diagram $f$ is continuous $\iff$ $f\circ q$ is continuous
$\xymatrix{ X \ar[d]_{q} \ar[dr]^{f\circ q} & \\ Y \ar[r]_f & Z }$

Let

$(X,\mathcal{ J })$ and

$(Y,\mathcal{ K })$ be topological spaces and let

$q:X\rightarrow Y$ be a quotient map. Then^[1]:

For any topological space, $(Z,\mathcal{ H })$ a map, $f:Y\rightarrow Z$ is continuous if and only if the composite map, $f\circ q$ , is continuous

Proof of claims

Notes

Jump up ↑ Recall that for an equivalence relation there is a natural map that sends each $x\in X$ to $[x]$ (the equivalence class containing $x$ ) which we denote here as $\pi:X\rightarrow\frac{X}{\sim}$ . Recall also that $\frac{X}{\sim}$ denotes the set of all equivalence classes of $\sim$ .

References

Jump up ↑ Introduction to Topological Manifolds - John M. Lee

OLD PAGE

Note: Motivation for quotient topology may be useful

Definition of the Quotient topology

$\begin{xy} \xymatrix{ X \ar[r]^p \ar[dr]_f & Q \ar@{.>}[d]^{\tilde{f}}\\ & Y } \end{xy}$

(OLD)Definition of Quotient topology

If $(X,\mathcal{J})$ is a topological space, $A$ is a set, and $p:(X,\mathcal{J})\rightarrow A$ is a surjective map then there exists exactly one topology $\mathcal{J}_Q$ relative to which $p$ is a quotient map. This is the quotient topology induced by $p$

[Expand]
The quotient topology is actually a topology

Quotient map

Let $(X,\mathcal{J})$ and $(Y,\mathcal{K})$ be topological spaces and let $p:X\rightarrow Y$ be a surjective map.

$p$ is a quotient map^[1] if we have $U\in\mathcal{K}\iff p^{-1}(U)\in\mathcal{J}$

That is to say $\mathcal{K}=\{V\in\mathcal{P}(Y)|p^{-1}(V)\in\mathcal{J}\}$

Also known as:

Identification map

Stronger than continuity

If we had $\mathcal{K}=\{\emptyset,Y\}$ then $p$ is automatically continuous (as it is surjective), the point is that $\mathcal{K}$ is the largest topology we can define on $Y$ such that $p$ is continuous

Theorems

[Expand]
Theorem: The quotient topology, $\mathcal{Q}$ is the largest topology such that the quotient map, $p$ , is continuous. That is to say any other topology such on $Y$ such that $p$ is continuous is contained in the quotient topology

For a map $p:X\rightarrow Y$ where $(X,\mathcal{J})$ is a Topological space we will show that the topology on $Y$ given by:

$\mathcal{Q}=\{V\in\mathcal{P}|p^{-1}(V)\in\mathcal{J}\}$

is the largest topology on $Y$ we can have such that $p$ is continuous

Proof method: suppose there's a larger topology, reach a contradiction.

Suppose that $\mathcal{K}$ is any topology on $Y$ and that $p:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$ is continuous.

Suppose that $\mathcal{K}\ne\mathcal{Q}$

Let $V\in\mathcal{K}$ such that $V\notin \mathcal{Q}$

By continuity of $p$ , $p^{-1}(V)\in\mathcal{J}$

This contradicts that $V\notin\mathcal{Q}$ as $\mathcal{Q}$ contains all subsets of $Y$ whose inverse image (preimage) is open in $X$

Thus any topology on $Y$ where $p$ is continuous is contained in the quotient topology

This theorem hints at the Characteristic property of the quotient topology

Quotient space

Given a Topological space $(X,\mathcal{J})$ and an Equivalence relation $\sim$ , then the map: $q:(X,\mathcal{J})\rightarrow(\tfrac{X}{\sim},\mathcal{Q})$ with $q:p\mapsto[p]$ (which is a quotient map) is continuous (as above)

The topological space $(\tfrac{X}{\sim},\mathcal{Q})$ is the quotient space^[2] where $\mathcal{Q}$ is the topology induced by the quotient

Also known as:

Identification space

References

Jump up ↑ Topology - Second Edition - James R Munkres
Jump up ↑ Introduction to topological manifolds - John M Lee - Second edition

[1] Jump up ↑ Recall that for an equivalence relation there is a natural map that sends each $x\in X$ to $[x]$ (the equivalence class containing $x$ ) which we denote here as $\pi:X\rightarrow\frac{X}{\sim}$ . Recall also that $\frac{X}{\sim}$ denotes the set of all equivalence classes of $\sim$ .

[ITTMJML-2] Jump up ↑ Introduction to Topological Manifolds - John M. Lee

[3] Jump up ↑ Topology - Second Edition - James R Munkres

[4] Jump up ↑ Introduction to topological manifolds - John M Lee - Second edition

[Note 1]

[1]

[1]

[2]

@@ Line 1: / Line 1: @@
-'''Note:''' [[Motivation for quotient topology]] may be useful
+{{Refactor notice|grade=A|msg=This page is an embarrassment}}
-{{Refactor notice}}
+==Definition==
+There are a few definitions of the quotient topology however they do not conflict. This page might change shape while things are put in place
+===[[Quotient topology/Equivalence relation definition|Quotient topology via an equivalence-relation definition]]===
+{{:Quotient topology/Equivalence relation definition}}
+===[[Quotient topology/Mapping to a set definition|Quotient topology via a mapping to a set definition]]===
+{{:Quotient topology/Mapping to a set definition}}
+: '''Claim 1: ''' these definitions are equivalent
+===[[Universal property of the quotient topology]]===
+{{:Universal property of the quotient topology/Statement}}
+==Proof of claims==
+==Notes==
+<references group="Note"/>
+==References==
+<references/>
+{{Topology navbox|plain}}
+{{Definition|Topology}}
+=OLD PAGE=
+'''Note:''' [[Motivation for quotient topology]] may be useful
 ==Definition of the Quotient topology==
 <math>

Difference between revisions of "Quotient topology"

Revision as of 14:20, 25 April 2016

Contents

Definition

Quotient topology via an equivalence-relation definition

Quotient topology via a mapping to a set definition

Universal property of the quotient topology

Proof of claims

Notes

References

OLD PAGE

Definition of the Quotient topology

(OLD)Definition of Quotient topology

Quotient map

Stronger than continuity

Theorems

Quotient space

See also

References

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